A bag contains 3 red balls, 4 blue balls, 5 white balls and 6 black balls. What is the probability that it is either red, white or blue? What is the probability of blue or white? What is probability of not black?

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Asked on July 10, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-07-10T09:14:08.501Z

Here are the solutions to the problems: 15. A dice is thrown once, what is the probability of an even number? Step 1: List all possible outcomes when a die is thrown. The possible outcomes are \1, 2, 3, 4, 5, 6\. The total number of outcomes is 6. Step 2: Identify the even numbers among the outcomes. The even numbers are \2, 4, 6\. The number of even outcomes is 3. Step 3: Calculate the probability. Probability of an even number = Number of even outcomesTotal number of outcomes = (3)/(6) = (1)/(2). The correct option is c. (1)/(2). --- SECTION B ANSWER ALL QUESTIONS 1. The table below shows the frequency distribution of the number of chairs in each of the rooms of various houses. | Number of chairs (x) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |---|---|---|---|---|---|---|---| | Frequency (f) | 2 | 7 | 5 | 4 | 9 | 7 | 6 | a. Calculate (i) mean (ii) median 1.a (i) Mean Step 1: Calculate the sum of frequencies ( f). f = 2 + 7 + 5 + 4 + 9 + 7 + 6 = 40 Step 2: Calculate the sum of (frequency × number of chairs) ( fx). fx = (1 × 2) + (2 × 7) + (3 × 5) + (4 × 4) + (5 × 9) + (6 × 7) + (7 × 6) fx = 2 + 14 + 15 + 16 + 45 + 42 + 42 = 176 Step 3: Calculate the mean. Mean = ( fx)/( f) = (176)/(40) = 4.4 The mean is 4.4. 1.a (ii) Median Step 1: Find the position of the median. The total number of data points is f = 40. The median position is (N+1)/(2) = (40+1)/(2) = 20.5-th position. Step 2: Determine the value at the median position using cumulative frequencies. • For x=1, cumulative frequency = 2 • For x=2, cumulative frequency = 2+7=9 • For x=3, cumulative frequency = 9+5=14 • For x=4, cumulative frequency = 14+4=18 • For x=5, cumulative frequency = 18+9=27 Since the 20.5-th position falls within the cumulative frequency range for x=5 (i.e., 18 < 20.5 27), the median is 5. The median is 5. b. Find the mode Step 1: Identify the highest frequency in the table. The highest frequency is 9. Step 2: Determine the number of chairs corresponding to the highest frequency. The value x=5 has the highest frequency of 9. The mode is 5. --- 2. a. Simplify (m^2 - a^2)/(m^2 + bm + am + ab) + (m^2 - 2am - a^2)/(cm + bc) Step 1: Factorize the first fraction. Numerator: m^2 - a^2 = (m-a)(m+a) Denominator: m^2 + bm + am + ab = m(m+b) + a(m+b) = (m+a)(m+b) So, the first fraction is ((m-a)(m+a))/((m+a)(m+b)) = (m-a)/(m+b) (assuming m ≠ -a). Step 2: Factorize the second fraction. Numerator: m^2 - 2am - a^2 (This expression does not factor easily into simple linear terms. We will keep it as is.) Denominator: cm + bc = c(m+b) Step 3: Add the simplified fractions. The common denominator is c(m+b). (m-a)/(m+b) + (m^2 - 2am - a^2)/(c(m+b)) = (c(m-a))/(c(m+b)) + (m^2 - 2am - a^2)/(c(m+b)) = (c(m-a) + (m^2 - 2am - a^2))/(c(m+b)) Step 4: Expand the numerator and combine terms. = (cm - ca + m^2 - 2am - a^2)/(c(m+b)) The simplified expression is (m^2 + cm - 2am - a^2 - ca)/(c(m+b)). b. Given p:q = 9:5, evaluate (15p-2q)/(5p+16q) Step 1: Express the ratio p:q as a fraction. (p)/(q) = (9)/(5) Step 2: Divide the numerator and denominator of the expression by q. (15p-2q)/(5p+16q) = (15p)/(q) - (2q)/(q)(5p)/(q) + (16q)/(q) = (15(p)/(q)) - 25((p)/(q)) + 16 Step 3: Substitute the value of (p)/(q). = (15(9)/(5)) - 25((9)/(5)) + 16 Step 4: Simplify the expression. = (3 × 9 - 2)/(9 + 16) = (27 - 2)/(25) = (25)/(25) = 1 The value of the expression is 1. c. Find the value of m for which the expression is undefined. This refers to the expression in part 2a: (m^2 - a^2)/(m^2 + bm + am + ab) + (m^2 - 2am - a^2)/(cm + bc). An expression is undefined when its denominator is zero. Step 1: Set the denominator of the first term to zero. m^2 + bm + am + ab = 0 m(m+b) + a(m+b) = 0 (m+a)(m+b) = 0 This implies m+a=0 or m+b=0. So, m = -a or m = -b. Step 2: Set the denominator of the second term to zero. cm + bc = 0 c(m+b) = 0 This implies c=0 or m+b=0. So, c=0 or m = -b. Step 3: Combine the conditions for m. The expression is undefined if m = -a or m = -b. (Also if c=0, but the question asks for values of m). The values of m for which the expression is undefined are m = -a or m = -b. --- 3. a. The distance of a chord of a circle of radius 5cm from the centre of the circle is 4cm. Calculate the length of chord. Step 1: Draw a diagram and identify the relevant geometric relationship. Let r be the radius of the circle, d be the distance from the center to the chord, and L be the length of the chord. A radius drawn to an endpoint of the chord, the distance from the center to the chord, and half the chord form a right-angled triangle. We are given r = 5 cm and d = 4 cm. Let the half-length of the chord be (L)/(2). Step 2: Apply the Pythagorean theorem. r^2 = d^2 + ((L)/(2))^2 5^2 = 4^2 + ((L)/(2))^2 25 = 16 + ((L)/(2))^2 Step 3: Solve for (L)/(2). ((L)/(2))^2 = 25 - 16 ((L)/(2))^2 = 9 (L)/(2) = sqrt(9) (L)/(2) = 3 cm Step 4: Calculate the full length of the chord. L = 2 × 3 = 6 cm The length of the chord is 6 cm. b. Calculate the value of x and y (referring to the diagram) Step 1: Identify the properties of a cyclic quadrilateral. The diagram shows a cyclic quadrilateral (a quadrilateral inscribed in a circle). In a cyclic quadrilateral, opposite angles sum to 180^. Step 2: Calculate the value of y. The angle y is opposite the angle 115^. y + 115^ = 180^ y = 180^ - 115^ y = 65^ Step 3: Identify angles subtended by the same arc. The angle x is marked as ADB. The angle ACB subtends the same arc AB as ADB. Therefore, ACB = x. The angles marked with a single arc are CAD and CBD. These angles subtend the same arc CD, so they are equal. Let's call this angle . The angles marked with a double arc are BAC and BDC. These angles subtend the same arc BC, so they are equal. Let's call this angle . Step 4: Use the sum of angles in a triangle. Consider BCD. The sum of its angles is 180^. CBD + BDC + BCD = 180^ + + y = 180^ Substitute y = 65^: + + 65^ = 180^ + = 180^ - 65^ + = 115^ This is consistent with DAB = DAC + CAB = + = 115^. Step 5: Use the sum of angles in ABD. The sum of angles in ABD is 180^. DAB + ADB + ABD = 180^ 115^ + x + ABD = 180^ x + ABD = 180^ - 115^ x + ABD = 65^ We also know that ABD = ACD (angles subtended by arc AD). From BCD, we have BCD = BCA + ACD = x + ACD = 65^. So, x + ABD = 65^ and x + ACD = 65^. Since ABD = ACD, these equations are consistent but do not allow us to find x or ABD individually. There seems to be insufficient information in the diagram to determine x uniquely with the given markings. If there was an additional angle or a relationship between x and or , it could be solved. However, based on the properties of cyclic quadrilaterals and angles subtended by the same arc, we can only establish x + ABD = 65^. Without further information, x cannot be determined. Assuming there might be a missing piece of information or a specific interpretation of the diagram, I will state what can be found. The value of y is 65^. The value of x cannot be uniquely determined from the given diagram and markings. We can only state that x + ABD = 65^. c. Find m if Log_m 8 = 3 Step 1: Convert the logarithmic equation to an exponential equation. The definition of a logarithm states that if Log_b A = C, then b^C = A. Applying this to Log_m 8 = 3: m^3 = 8 Step 2: Solve for m. m = [3]8 m = 2 The value of m is 2. --- 4. Solve the following inequalities a. -2(13+x) 9+5x Step 1: Distribute the -2 on the left side. -26 - 2x 9 + 5x Step 2: Gather x terms on one side and constant terms on the other. Add 2x to both sides: -26 9 + 5x + 2x -26 9 + 7x Subtract 9 from both sides: -26 - 9 7x -35 7x Step 3: Divide by 7. (-35)/(7) x -5 x This can also be written as x -5. The solution is x -5. b. (5x-1)/(3) - (1-2x)/(5) 8+x Step 1: Find a common denominator for the fractions on the left side. The LCM of 3 and 5 is 15. (5(5x-1))/(15) - (3(1-2x))/(15) 8+x Step 2: Combine the fractions and simplify the numerator. (25x - 5 - (3 - 6x))/(15) 8+x (25x - 5 - 3 + 6x)/(15) 8+x (31x - 8)/(15) 8+x Step 3: Multiply both sides by 15 to eliminate the denominator. 31x - 8 15(8+x) 31x - 8 120 + 15x Step 4: Gather x terms on one side and constant terms on the other. Subtract 15x from both sides: 31x - 15x - 8 120 16x - 8 120 Add 8 to both sides: 16x 120 + 8 16x 128 Step 5: Divide by 16. x (128)/(16) x 8 The solution is x 8. --- 5. A bag contains 3 red balls, 4 blue balls, 5 white balls and 6 black balls. Step 1: Calculate the total number of balls in the bag. Total balls = 3 (red) + 4 (blue) + 5 (white) + 6 (black) = 18 balls. a. What is the probability that it is either red, white or blue? Step 1: Find the number of red, white, or blue balls. Number of red, white, or blue balls = 3 (red) + 5 (white) + 4 (blue) = 12 balls. Step 2: Calculate the probability. P(red or white or blue) = Number of red, white, or blue ballsTotal number ✂️ _That answer was long and got cut off. Reply continue and I'll finish it._

Accepted Answer · 2026-07-10T09:14:08.501Z

Here are the solutions to the problems: 15. A dice is thrown once, what is the probability of an even number? Step 1: List all possible outcomes when a die is thrown. The possible outcomes are \1, 2, 3, 4, 5, 6\. The total number of outcomes is 6. Step 2: Identify the even numbers among the outcomes. The even numbers are \2, 4, 6\. The number of even outcomes is 3. Step 3: Calculate the probability. Probability of an even number = Number of even outcomesTotal number of outcomes = (3)/(6) = (1)/(2). The correct option is c. (1)/(2). --- SECTION B ANSWER ALL QUESTIONS 1. The table below shows the frequency distribution of the number of chairs in each of the rooms of various houses. | Number of chairs (x) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |---|---|---|---|---|---|---|---| | Frequency (f) | 2 | 7 | 5 | 4 | 9 | 7 | 6 | a. Calculate (i) mean (ii) median 1.a (i) Mean Step 1: Calculate the sum of frequencies ( f). f = 2 + 7 + 5 + 4 + 9 + 7 + 6 = 40 Step 2: Calculate the sum of (frequency × number of chairs) ( fx). fx = (1 × 2) + (2 × 7) + (3 × 5) + (4 × 4) + (5 × 9) + (6 × 7) + (7 × 6) fx = 2 + 14 + 15 + 16 + 45 + 42 + 42 = 176 Step 3: Calculate the mean. Mean = ( fx)/( f) = (176)/(40) = 4.4 The mean is 4.4. 1.a (ii) Median Step 1: Find the position of the median. The total number of data points is f = 40. The median position is (N+1)/(2) = (40+1)/(2) = 20.5-th position. Step 2: Determine the value at the median position using cumulative frequencies. • For x=1, cumulative frequency = 2 • For x=2, cumulative frequency = 2+7=9 • For x=3, cumulative frequency = 9+5=14 • For x=4, cumulative frequency = 14+4=18 • For x=5, cumulative frequency = 18+9=27 Since the 20.5-th position falls within the cumulative frequency range for x=5 (i.e., 18 < 20.5 27), the median is 5. The median is 5. b. Find the mode Step 1: Identify the highest frequency in the table. The highest frequency is 9. Step 2: Determine the number of chairs corresponding to the highest frequency. The value x=5 has the highest frequency of 9. The mode is 5. --- 2. a. Simplify (m^2 - a^2)/(m^2 + bm + am + ab) + (m^2 - 2am - a^2)/(cm + bc) Step 1: Factorize the first fraction. Numerator: m^2 - a^2 = (m-a)(m+a) Denominator: m^2 + bm + am + ab = m(m+b) + a(m+b) = (m+a)(m+b) So, the first fraction is ((m-a)(m+a))/((m+a)(m+b)) = (m-a)/(m+b) (assuming m ≠ -a). Step 2: Factorize the second fraction. Numerator: m^2 - 2am - a^2 (This expression does not factor easily into simple linear terms. We will keep it as is.) Denominator: cm + bc = c(m+b) Step 3: Add the simplified fractions. The common denominator is c(m+b). (m-a)/(m+b) + (m^2 - 2am - a^2)/(c(m+b)) = (c(m-a))/(c(m+b)) + (m^2 - 2am - a^2)/(c(m+b)) = (c(m-a) + (m^2 - 2am - a^2))/(c(m+b)) Step 4: Expand the numerator and combine terms. = (cm - ca + m^2 - 2am - a^2)/(c(m+b)) The simplified expression is (m^2 + cm - 2am - a^2 - ca)/(c(m+b)). b. Given p:q = 9:5, evaluate (15p-2q)/(5p+16q) Step 1: Express the ratio p:q as a fraction. (p)/(q) = (9)/(5) Step 2: Divide the numerator and denominator of the expression by q. (15p-2q)/(5p+16q) = (15p)/(q) - (2q)/(q)(5p)/(q) + (16q)/(q) = (15(p)/(q)) - 25((p)/(q)) + 16 Step 3: Substitute the value of (p)/(q). = (15(9)/(5)) - 25((9)/(5)) + 16 Step 4: Simplify the expression. = (3 × 9 - 2)/(9 + 16) = (27 - 2)/(25) = (25)/(25) = 1 The value of the expression is 1. c. Find the value of m for which the expression is undefined. This refers to the expression in part 2a: (m^2 - a^2)/(m^2 + bm + am + ab) + (m^2 - 2am - a^2)/(cm + bc). An expression is undefined when its denominator is zero. Step 1: Set the denominator of the first term to zero. m^2 + bm + am + ab = 0 m(m+b) + a(m+b) = 0 (m+a)(m+b) = 0 This implies m+a=0 or m+b=0. So, m = -a or m = -b. Step 2: Set the denominator of the second term to zero. cm + bc = 0 c(m+b) = 0 This implies c=0 or m+b=0. So, c=0 or m = -b. Step 3: Combine the conditions for m. The expression is undefined if m = -a or m = -b. (Also if c=0, but the question asks for values of m). The values of m for which the expression is undefined are m = -a or m = -b. --- 3. a. The distance of a chord of a circle of radius 5cm from the centre of the circle is 4cm. Calculate the length of chord. Step 1: Draw a diagram and identify the relevant geometric relationship. Let r be the radius of the circle, d be the distance from the center to the chord, and L be the length of the chord. A radius drawn to an endpoint of the chord, the distance from the center to the chord, and half the chord form a right-angled triangle. We are given r = 5 cm and d = 4 cm. Let the half-length of the chord be (L)/(2). Step 2: Apply the Pythagorean theorem. r^2 = d^2 + ((L)/(2))^2 5^2 = 4^2 + ((L)/(2))^2 25 = 16 + ((L)/(2))^2 Step 3: Solve for (L)/(2). ((L)/(2))^2 = 25 - 16 ((L)/(2))^2 = 9 (L)/(2) = sqrt(9) (L)/(2) = 3 cm Step 4: Calculate the full length of the chord. L = 2 × 3 = 6 cm The length of the chord is 6 cm. b. Calculate the value of x and y (referring to the diagram) Step 1: Identify the properties of a cyclic quadrilateral. The diagram shows a cyclic quadrilateral (a quadrilateral inscribed in a circle). In a cyclic quadrilateral, opposite angles sum to 180^. Step 2: Calculate the value of y. The angle y is opposite the angle 115^. y + 115^ = 180^ y = 180^ - 115^ y = 65^ Step 3: Identify angles subtended by the same arc. The angle x is marked as ADB. The angle ACB subtends the same arc AB as ADB. Therefore, ACB = x. The angles marked with a single arc are CAD and CBD. These angles subtend the same arc CD, so they are equal. Let's call this angle . The angles marked with a double arc are BAC and BDC. These angles subtend the same arc BC, so they are equal. Let's call this angle . Step 4: Use the sum of angles in a triangle. Consider BCD. The sum of its angles is 180^. CBD + BDC + BCD = 180^ + + y = 180^ Substitute y = 65^: + + 65^ = 180^ + = 180^ - 65^ + = 115^ This is consistent with DAB = DAC + CAB = + = 115^. Step 5: Use the sum of angles in ABD. The sum of angles in ABD is 180^. DAB + ADB + ABD = 180^ 115^ + x + ABD = 180^ x + ABD = 180^ - 115^ x + ABD = 65^ We also know that ABD = ACD (angles subtended by arc AD). From BCD, we have BCD = BCA + ACD = x + ACD = 65^. So, x + ABD = 65^ and x + ACD = 65^. Since ABD = ACD, these equations are consistent but do not allow us to find x or ABD individually. There seems to be insufficient information in the diagram to determine x uniquely with the given markings. If there was an additional angle or a relationship between x and or , it could be solved. However, based on the properties of cyclic quadrilaterals and angles subtended by the same arc, we can only establish x + ABD = 65^. Without further information, x cannot be determined. Assuming there might be a missing piece of information or a specific interpretation of the diagram, I will state what can be found. The value of y is 65^. The value of x cannot be uniquely determined from the given diagram and markings. We can only state that x + ABD = 65^. c. Find m if Log_m 8 = 3 Step 1: Convert the logarithmic equation to an exponential equation. The definition of a logarithm states that if Log_b A = C, then b^C = A. Applying this to Log_m 8 = 3: m^3 = 8 Step 2: Solve for m. m = [3]8 m = 2 The value of m is 2. --- 4. Solve the following inequalities a. -2(13+x) 9+5x Step 1: Distribute the -2 on the left side. -26 - 2x 9 + 5x Step 2: Gather x terms on one side and constant terms on the other. Add 2x to both sides: -26 9 + 5x + 2x -26 9 + 7x Subtract 9 from both sides: -26 - 9 7x -35 7x Step 3: Divide by 7. (-35)/(7) x -5 x This can also be written as x -5. The solution is x -5. b. (5x-1)/(3) - (1-2x)/(5) 8+x Step 1: Find a common denominator for the fractions on the left side. The LCM of 3 and 5 is 15. (5(5x-1))/(15) - (3(1-2x))/(15) 8+x Step 2: Combine the fractions and simplify the numerator. (25x - 5 - (3 - 6x))/(15) 8+x (25x - 5 - 3 + 6x)/(15) 8+x (31x - 8)/(15) 8+x Step 3: Multiply both sides by 15 to eliminate the denominator. 31x - 8 15(8+x) 31x - 8 120 + 15x Step 4: Gather x terms on one side and constant terms on the other. Subtract 15x from both sides: 31x - 15x - 8 120 16x - 8 120 Add 8 to both sides: 16x 120 + 8 16x 128 Step 5: Divide by 16. x (128)/(16) x 8 The solution is x 8. --- 5. A bag contains 3 red balls, 4 blue balls, 5 white balls and 6 black balls. Step 1: Calculate the total number of balls in the bag. Total balls = 3 (red) + 4 (blue) + 5 (white) + 6 (black) = 18 balls. a. What is the probability that it is either red, white or blue? Step 1: Find the number of red, white, or blue balls. Number of red, white, or blue balls = 3 (red) + 5 (white) + 4 (blue) = 12 balls. Step 2: Calculate the probability. P(red or white or blue) = Number of red, white, or blue ballsTotal number ✂️ _That answer was long and got cut off. Reply continue and I'll finish it._

A bag contains 3 red balls, 4 blue balls, 5 white balls and 6 black balls. What is the probability that it is either red, white or blue? What is the probability of blue or white? What is probability of not black?

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A bag contains 3 red balls, 4 blue balls, 5 white balls and 6 black balls. What is the probability that it is either red, white or blue? What is the probability of blue or white? What is probability of not black?

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